Hardy–Weinberg Equilibrium

Historical Context and Significance

The Hardy–Weinberg equilibrium is a foundational principle in population genetics, describing the expected distribution of genotypes in a population under certain conditions. It was independently derived by mathematician Godfrey Hardy and physician Wilhelm Weinberg in 1908, addressing questions about why dominant alleles do not necessarily become more common over time. The equilibrium provides a baseline for understanding evolutionary change and the effects of various evolutionary forces.

Genotype and Allele Frequencies

Genotype Frequencies

Genotype frequency is the proportion of individuals in a population with a specific genotype. For a locus with two alleles, A and a, the possible genotypes are AA, Aa, and aa.

• Genotype Frequency Calculation: Divide the number of individuals with each genotype by the total population size.

• Example: In a population of 150 individuals: AA = 54, Aa = 72, aa = 24. - AA: - Aa: - aa: - Sum:

Allele Frequencies

Allele frequency is the proportion of a specific allele among all alleles at a locus in the population. Each individual has two alleles per locus.

• Allele Counting Method: Count alleles from each genotype (homozygotes contribute two, heterozygotes one).

• Example: Using the above population: - Number of A alleles: - Number of a alleles: - Total alleles: - Frequency of A: - Frequency of a:

• Symbols: Conventionally, p = frequency of A, q = frequency of a.

• Check:

Allele Frequency Calculation Table

Alternative Method: Using Genotype Frequencies

• Formula: Where , , are genotype frequencies.

• Example: ;

More Than Two Alleles

For loci with more than two alleles, count each allele across all genotypes. For three alleles (A, B, C), frequencies are labeled p, q, r.

• Frequencies: , ,

Hardy–Weinberg Equilibrium: Definition and Equations

Principle and Equations

Hardy–Weinberg equilibrium describes the expected genotype frequencies in a population given allele frequencies, assuming certain conditions are met.

• For two alleles (A and a): - AA: - Aa: - aa: - -

• Example: If , : - AA: - Aa: - aa:

Equilibrium Meaning

• Under Hardy–Weinberg equilibrium, allele and genotype frequencies remain constant from generation to generation.

• If a population is not initially at equilibrium, it will reach equilibrium in one generation under random mating.

• Equilibrium: , (allele frequencies in offspring equal those in parents).

Assumptions of Hardy–Weinberg Equilibrium

Key Assumptions

• Random mating (no inbreeding or assortative mating)

• No mutation (alleles do not change)

• No genetic drift (population is infinitely large)

• No natural selection (all genotypes have equal survival and reproduction)

• No gene flow (population is closed to migration)

Violation of these assumptions leads to changes in allele and/or genotype frequencies, i.e., evolution.

Applications of Hardy–Weinberg Equilibrium

Detecting Deviations

• Compare observed genotype numbers to expected numbers under equilibrium.

• Use statistical tests (e.g., chi-square) to determine if deviations are significant.

• Chi-square formula: Where O = observed, E = expected.

• Degrees of freedom: Where n = number of classes, k = number of independent parameters.

Dominant Alleles and Phenotypes

• When alleles are not codominant, allele counting is not possible.

• Assume equilibrium to estimate allele frequencies from phenotype frequencies.

• Example: For a recessive phenotype frequency , estimate , then .

Extensions of Hardy–Weinberg Equilibrium

Linkage Disequilibrium

Linkage disequilibrium refers to the nonrandom association of alleles at different loci.

• D: Linkage disequilibrium parameter; D = 0 indicates equilibrium.

• Recombination reduces D over generations.

More Than Two Alleles

• For three alleles (A, B, C) with frequencies p, q, r: - AA: - AB: - AC: - BB: - BC: - CC:

X-Linked Genes

• Females (XX): Hardy–Weinberg proportions apply (e.g., AA = , Aa = , aa = ).

• Males (XY): Only one X chromosome; genotype frequencies equal allele frequencies (A = p, a = q).

• Example: For Xg blood group, Xga = 0.659, Xg = 0.341. - Female Xg(a−): - Male Xg(a−):

Hardy–Weinberg Equilibrium and Evolution

Evolutionary Forces

• Mutation: Random changes introduce new alleles.

• Natural Selection: Differential survival/reproduction alters allele frequencies.

• Genetic Drift: Random fluctuations, especially in small populations.

• Gene Flow: Migration introduces new alleles.

Hardy–Weinberg equilibrium provides a baseline; deviations indicate evolutionary processes.

Appendix: Proofs and Statistical Tests

Allele Frequency from Genotype Frequency

• Proof: For two alleles, A and a, with genotype counts NAA, NAa, Naa: Where N = total individuals.

Chi-Square Test for Equilibrium

• Formula:

• Degrees of freedom:

• Interpretation: If is less than the critical value, accept equilibrium; otherwise, reject.

Summary

• Hardy–Weinberg equilibrium is a mathematical model describing genotype and allele frequencies in populations.

• It provides a baseline for detecting evolutionary change.

• Assumptions include random mating, no mutation, no genetic drift, no selection, and no gene flow.

• Deviations from equilibrium indicate the action of evolutionary forces.

Additional info: The notes include expanded explanations, examples, and formulas to ensure completeness and academic quality for Genetics students.